Finite Group Databasebeta

← #29716692

Subgroups — #29716692

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451626C212yesmaximal, complemented
2#21448473C211yesmaximal, characteristic
3#21447171C212yesmaximal, complemented
4#21446982C212yesmaximal, complemented
5#21446805C212yesmaximal, complemented
6#21446227C212yesmaximal
7#21446100C212yesmaximal, complemented
8#21445807C211yesmaximal, characteristic, complemented
9C61×#3501C211yesmaximal, characteristic, complemented
10C2×#14519745C2211yescharacteristic
11C22×#11758198C2211yescharacteristic
12C2×#14519692C2212yes—
13#16839697—22no—
14#16839684—24no—
15#16839673—22no—
16#16839662—21no—
17#16839653—21no—
18#16839652—21no—
19#16839642C2211yescharacteristic
20#16839641—21no—
21#16839634C2211yescharacteristic
22#16839634C2214yes—
23#16839622C2212yes—
24#16839617C2211yescharacteristic
25#16839617C2214yes—
26C4×C22⋊Dic61C2211yescharacteristic
27#16839529C2212yes—
28#16839529C2214yes—
29#16839529C2214yes—
30C61×#554—24no—
31C61×#554—22no—
32D4×Q8×C61—22no—
33C61×#544—21no—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C24×D61C2311yescharacteristic
42C22×C22⋊D61—21no—
43C23×Dic61C2311yescharacteristic
44#14519745—22no—
45#14519745Q814yescomplemented
46C2×C4⋊Dic61—22no—
47C2×#11758196—24no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61C2314yes—
51C2×C22⋊Dic61C2311yescharacteristic
52C2×C22⋊Dic61C2311yescharacteristic
53C2×#11758198—21no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—24no—
59C2×#11758198—24no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×C4×Dic61—21no—
63#14519692—24no—
64#14519684—24no—
65#14519684—22no—
66#14519684—21no—
67#14519684—21no—
68#14519684—21no—
69C22×Q8×C61—21no—
70C22×Q8×C61D412yescomplemented
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—24no—
77#125×C61—22no—
78#125×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—24no—
97#117×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D414yescomplemented
102C2×C4⋊C4×C61D412yescomplemented
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61C2314yes—
107C2×C42×C61D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C24×C61C2411yescharacteristic
110C23×D61—21no—
111C23×D61—22no—
112C23×D61—21no—
113C23×D61—22no—
114C23×D61—81no—
115C23×D61—81no—
116C23×D61C2×Q812yescomplemented
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—82no—
122C2×Q8×C61—82no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61C2×Q812yes—
126C2×D4×C61#4112yes—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—81no—
132C22×C4×C61—81no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D414yes—
137C22×C4×C61C2×D414yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—81no—
144C2×C22⋊D61—81no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—24no—
150C22×Dic61—81no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C42×C61—24no—
168C42×C61—21no—
169C42×C61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—24no—
172C22⋊Dic61—22no—
173C22⋊Dic61—24no—
174C22⋊Dic61—81no—
175C22⋊Dic61C2×Q812yescomplemented
176#11758198—24no—
177#11758198—41no—
178#11758198—41no—
179#11758198—42no—
180#11758198—81no—
181#11758198—84no—
182#11758198—84no—
183#11758198—81no—
184C4⋊Dic61—82no—
185#11758196—81no—
186#11758196—84no—
187C4×Dic61—81no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61C22⋊Q812yes—
195C23×C61C22×D411yescharacteristic
196C23×C61C22×D412yes—
197C23×C61C22×Q811yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C612−1+412yes—
200C22×D61—22no—
201C22×D61—24no—
202C22×D61—44no—
203C22×D61—41no—
204C22×D61—44no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—84no—
214Q8×C61—44no—
215Q8×C61—82no—
216Q8×C61—82no—
217Q8×C61—82no—
218D4×C61—24no—
219D4×C61—24no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—44no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—44no—
235C2×C4×C61—81no—
236C2×C4×C61—82no—
237C2×C4×C61—82no—
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12012yes—
241C2×C4×C61C22⋊Q812yes—
242C22⋊D61—84no—
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—44no—
248C2×Dic61—44no—
249C2×Dic61—41no—
250C2×Dic61—44no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—44no—
256#24352—611nomaximal, Sylow
257C22×C61—22no—
258C22×C61—21no—
259C22×C61—22no—
260C22×C61—21no—
261C22×C61—22no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—81no—
269C22×C61C2×#12011yescharacteristic
270C22×C61C2×C22⋊Q811yescharacteristic
271C22×C61#53611yescharacteristic
272C22×C61#54812yes—
273C22×C61D4×Q812yes—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—48no—
277C2×D61—42no—
278C2×D61—84no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—22no—
287C4×C61—41no—
288C4×C61—44no—
289C4×C61—44no—
290C4×C61—41no—
291Dic61—42no—
292Dic61—84no—
293Dic61—81no—
294Dic61—81no—
295#3501D6111yescharacteristic, complemented
296#2626—612no—
297#2321—611no—
298#2062—612no—
299#2029—612no—
300#1996—612no—
301#1889—612no—
302#1867—612no—
303#1799—611no—
304C2×C61—21no—
305C2×C61—22no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—42no—
309C2×C61#240911yescharacteristic
310C2×C61#302611yescharacteristic
311C2×C61#302711yescharacteristic
312D61—82no—
313#554—24no—
314#554—22no—
315D4×Q8—22no—
316#544—21no—
317#543—21no—
318#543C2×D6112yes—
319#533—21no—
320C2×C22⋊Q8—21no—
321C2×C22⋊Q8C2×D6112yes—
322C2×#120—611no—
323#518C2×D6112yes—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C22⋊C4—611no—
326C2×#99—612no—
327#399—1222no—
328#396—1224no—
329#393—1222no—
330#390—1221no—
331#388—1221no—
332#388—1221no—
333#386—611no—
334#386—1221no—
335#385—611no—
336#385—614no—
337#380—612no—
338#379—611no—
339#379—614no—
340C4×C22⋊C4—611no—
341#342—612no—
342#342—614no—
343#342—614no—
344C61#2435211yescharacteristic, complemented, Sylow
345C25—611no—
346C22×Q8—21no—
347C22×Q8C22⋊D6112yescomplemented
348C22×D4—1221no—
349C22×D4C22×D6111yescharacteristic
350C23×C4—611no—
351C23×C4C22⋊D6111yescharacteristic
352C23×C4C22×D6111yescharacteristic
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—24no—
356#125—22no—
357#125—24no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370#120—614no—
371#120—1222no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—24no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1224no—
384C2×C4⋊C4C22⋊D6111yescharacteristic
385C2×C4⋊C4C22⋊D6114yescomplemented
386C2×C4⋊C4C22⋊D6112yescomplemented
387C2×C22⋊C4—22no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—614no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4C22⋊D6111yescharacteristic
402C2×C22⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4C22×D6114yes—
404C2×C42—1221no—
405C2×C42C22⋊D6111yescharacteristic
406#99—1224no—
407#95—1221no—
408#95—1222no—
409#95—1221no—
410#95—1224no—
411#95—1221no—
412C24—612no—
413C24—1221no—
414C24—1222no—
415C24—1222no—
416C24—1221no—
417C24—4881no—
418C24—4881no—
419C24C2×C22⋊D6111yescharacteristic
420C24C23×D6111yescharacteristic
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—1224no—
430C2×D4—4881no—
431C2×D4—4881no—
432C2×D4Q8×D6112yes—
433C2×D4#1175822612yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1224no—
443C22×C4—1221no—
444C22×C4—1222no—
445C22×C4—4881no—
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6114yes—
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6114yes—
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6112yes—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6112yes—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—4884no—
467C4⋊C4—4881no—
468C4⋊C4—4882no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—612no—
475C22⋊C4—1221no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—1222no—
479C22⋊C4—1224no—
480C22⋊C4—4881no—
481C22⋊C4—4884no—
482C22⋊C4—4881no—
483C22⋊C4—4884no—
484C22⋊C4—4881no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C22⋊C4—2442no—
488C42—24no—
489C42—21no—
490C42—21no—
491C42—4881no—
492C23—21no—
493C23—21no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—1222no—
499C23—1224no—
500C23—4882no—
501C23—4884no—
502C23—4882no—
503C23—2444no—
504C23—2441no—
505C23—2444no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2442no—
512C23#1451975312yes—
513C23C22×C22⋊D6111yescharacteristic
514C23C22×C22⋊D6112yes—
515C23C2×Q8×D6111yescharacteristic
516C23C2×#1175822611yescharacteristic
517C23#1451982512yes—
518Q8—44no—
519Q8—82no—
520Q8—82no—
521Q8—82no—
522D4—24no—
523D4—24no—
524D4—4884no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—1222no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2441no—
549C2×C4—2444no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2444no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#1451974611yescharacteristic
557C2×C4#1451974611yescharacteristic
558C2×C4#1451974612yes—
559C2×C4#1451975312yes—
560C22—22no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—81no—
572C22—4884no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—2442no—
578C22—2442no—
579C22—2448no—
580C22—2442no—
581C22C2×#1451974611yescharacteristic
582C22C2×#1451975311yescharacteristic
583C22#1684033611yescharacteristic
584C22#1684042312yes—
585C22Q8×C22⋊D6112yes—
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—44no—
592C4—44no—
593C4—41no—
594C4—4884no—
595C4—4881no—
596C4—4881no—
597C4—2442no—
598C2—22no—
599C2—21no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—4882no—
604C2#2144917811yescharacteristic
605C2#2145521611yescharacteristic
606C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC2×#14519745C22
hypercenter#3501D61
hypocenterC61#24352
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21451626C2semidirect
3. #21447171C2semidirect
4. #21446982C2semidirect
5. #21446805C2semidirect
7. #21446100C2semidirect
8. #21445807C2semidirect
9. C61×#3501C2semidirect
45. #14519745Q8semidirect
70. C22×Q8×C61D4semidirect
101. C2×C4⋊C4×C61D4semidirect
102. C2×C4⋊C4×C61D4semidirect
116. C23×D61C2×Q8semidirect
175. C22⋊Dic61C2×Q8semidirect
295. #3501D61semidirect
344. C61#24352semidirect
347. C22×Q8C22⋊D61semidirect
385. C2×C4⋊C4C22⋊D61semidirect
386. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214516262
#214484732
#214471712
#214469822
#214468052
#214462272
#214461002
#214458072
C61×#35012
#2435261